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A point charge q moves unifromlyand rectilnearly with arelativisticwith a relativisticvelocityof light(beta = v//c). Find theelectric fieldstrengthE produced by the charge at the pointwhose radius vectorrelatives to the charge is equal to rand formsan angle theta with itsvelocity vector. |
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Answer» Solution :Suppose the charge `Q` moves in thepositivedirectionof the x-axis of the frame `K`. Let us GO over TOTHE moving frame `K'`, at whose origin the charge is at rest. We take the `x`and `x'` axes of the twoframesto becoincident,and the`y & y'` axes, to beparallel. In the `K'` frame, `vec(E) = (1)/(4pi epsilon_(0)) (q vec(r))/(r^(3))`, and this has the followingcomponents, `E'_(x) = (1)/(4pi epsilon_(0)) (q x')/(r'^(3)), E'_(y) = (1)/(4pi epsilon_(0)) (q y')/(r'^(3))` Now let us go BACK to the frame`K`. At the moment, whenthe originsof the two frames coincide, we take `t = 0`. Then, `x = r cos theta = x' sqrt(1 - (v^(2))/(c^(2))), y = r sin theta = y'` Also, `E'_(x) = E'_(x), E_(y) = E'_(y)//sqrt(1 - v^(2)//c^(2))` Form these equations, `r'^(2) = (r^(2)(1 - beta^(2) sin^(2) theta))/(1 - beta^(2))` `vec(E) = (q)/(4pi epsilon_(0)) (1)/(r^(3) (1 - beta^(2) sin^(2) theta)^(3//2)) [ (1 - beta^(2))^(3//2)(x' hat(i) + (y')/(sqrt(1 - beta^(2)))hat(j))]` `= (q vec(r) (1 - beta^(2)))/(4pi epsilon_(0) r^(3) (1 - beta^(2) sin^(2) theta)^(3//2))` |
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